A103941 Number of unrooted loopless n-edge maps in the plane (planar with a distinguished outside face).

1, 1, 2, 6, 22, 103, 614, 3872, 26414, 186988, 1367976, 10254326, 78461338, 610598818, 4821248244, 38546510368, 311560875422, 2542507084588, 20925300483992, 173530381632724, 1448900079476152, 12172334379246523, 102833593763830038, 873187910184763024, 7449120536014301138

Offset

0,3

References

V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..500

V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.

Formula

For n > 0, a(n) = (1/(2n))*[binomial(4n, n)/(3n+1) + Sum_{0<k<n, k|n} phi(n/k)*binomial(4k, k)+q(n)] where phi is the Euler function A000010, q(n)=0 if n is even and q(n)=binomial(2n, (n-1)/2) if n is odd.

Mathematica

a[n_] := (1/(2n)) (Binomial[4n, n]/(3n+1) + Sum[Boole[0 < k < n] EulerPhi[ n/k] Binomial[4k, k], {k, Divisors[n]}] + q[n]);
q[n_] := If[EvenQ[n], 0, Binomial[2n, (n-1)/2]];
Array[a, 20] (* Jean-François Alcover, Sep 01 2019 *)

Prog

(PARI) a(n) = {if(n==0, 1, (sumdiv(n, d, if(d<n, 1, 1/(3*n+1)) * eulerphi(n/d) * binomial(4*d, d)) + if(n%2, binomial(2*n, (n-1)/2)))/(2*n))} \\ Andrew Howroyd, Mar 28 2021

Crossrefs

Cf. A002293, A006390, A103942, A000260, A005470.

Sequence in context: A345194 A251181 A309579 * A064643 A218531 A339280

Adjacent sequences: A103938 A103939 A103940 * A103942 A103943 A103944

Keyword

easy,nonn

Author

Valery A. Liskovets, Mar 17 2005

Extensions

a(0)=1 prepended and terms a(21) and beyond from Andrew Howroyd, Mar 28 2021

Status

approved